New approach to the calculation ofF1(α)in massless quantum electrodynamics

$F_1\left(\alpha \right)$ is defined as the contribution of the one-fermion-loop diagrams to the divergent part of the photon propagator in massless quantum electrodynamics. To sixth order, the perturbation expansion of $F_1\left(\alpha \right)$ has rational coefficients: $F_1\left(\alpha \right)=\left(\frac{2}{3}\right)\left(\frac{\alpha }{2\pi }\right)+{\left(\frac{\alpha }{2\pi }\right)}^2-\left(\frac{1}{4}\right){(\frac{\alpha }{2\pi }}^3+\cdots$. It is not known whether the next term in this series is a rational number; however, we propose a new method, which uses integration by parts, for evaluating Feynman integrals which give rational numbers. Using this method we easily rederive the first three terms in the series for $F_1\left(\alpha \right)$ and three other two-loop integrals, including the fourth-order correction to the vertex function ${\Gamma }^{\mu }\mathrm{}(p,p)\mathrm{}$. We believe that our new integration techniques are powerful enough to evaluate the fourth term in the series for $F_1\left(\alpha \right)$ if it is a rational number.